An angle in standard position has its terminal side passing through the point $\left(3,,,4\right)$ in the coordinate plane. Estimate the measure of this angle within $10°$.

• A. $75°$
• B. $30°$
• C. $53°$
• D. $120°$

Answer: (C)

Verified step by step guidance

1. Identify the coordinates of the point through which the terminal side of the angle passes. Here, the point is (3, 4).
2. Recall that the angle in standard position is measured from the positive x-axis to the terminal side. To find this angle, we can use the tangent function, which relates the y-coordinate and x-coordinate of the point: \(\tan(\theta) = \frac{y}{x}\).
3. Calculate the ratio \(\frac{y}{x}\) using the given point: \(\frac{4}{3}\).
4. Use the inverse tangent function to find the angle \(\theta\): \(\theta = \tan^{-1}\left(\frac{4}{3}\right)\).
5. Estimate the value of \(\theta\) in degrees and compare it to the given options to find the closest angle measure.